MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  distrlem5pr Unicode version

Theorem distrlem5pr 8667
Description: Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
distrlem5pr  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A  .P.  B
)  +P.  ( A  .P.  C ) )  C_  ( A  .P.  ( B  +P.  C ) ) )

Proof of Theorem distrlem5pr
Dummy variables  x  y  z  w  v  u  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulclpr 8660 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
213adant3 975 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
3 mulclpr 8660 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
433adant2 974 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
5 df-plp 8623 . . . . 5  |-  +P.  =  ( x  e.  P. ,  y  e.  P.  |->  { f  |  E. g  e.  x  E. h  e.  y  f  =  ( g  +Q  h ) } )
6 addclnq 8585 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
75, 6genpelv 8640 . . . 4  |-  ( ( ( A  .P.  B
)  e.  P.  /\  ( A  .P.  C )  e.  P. )  -> 
( w  e.  ( ( A  .P.  B
)  +P.  ( A  .P.  C ) )  <->  E. v  e.  ( A  .P.  B
) E. u  e.  ( A  .P.  C
) w  =  ( v  +Q  u ) ) )
82, 4, 7syl2anc 642 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( ( A  .P.  B )  +P.  ( A  .P.  C ) )  <->  E. v  e.  ( A  .P.  B
) E. u  e.  ( A  .P.  C
) w  =  ( v  +Q  u ) ) )
9 df-mp 8624 . . . . . . . 8  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. g  e.  w  E. h  e.  v  x  =  ( g  .Q  h ) } )
10 mulclnq 8587 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
119, 10genpelv 8640 . . . . . . 7  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( u  e.  ( A  .P.  C )  <->  E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z ) ) )
12113adant2 974 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
u  e.  ( A  .P.  C )  <->  E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z
) ) )
1312anbi2d 684 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( A  .P.  B )  /\  u  e.  ( A  .P.  C ) )  <->  ( v  e.  ( A  .P.  B
)  /\  E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z
) ) ) )
14 df-mp 8624 . . . . . . . . 9  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { f  |  E. g  e.  w  E. h  e.  v  f  =  ( g  .Q  h ) } )
1514, 10genpelv 8640 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( v  e.  ( A  .P.  B )  <->  E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y ) ) )
16153adant3 975 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
v  e.  ( A  .P.  B )  <->  E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y
) ) )
17 distrlem4pr 8666 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  A  /\  y  e.  B )  /\  (
f  e.  A  /\  z  e.  C )
) )  ->  (
( x  .Q  y
)  +Q  ( f  .Q  z ) )  e.  ( A  .P.  ( B  +P.  C ) ) )
18 oveq12 5883 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( v  +Q  u
)  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) )
1918eqeq2d 2307 . . . . . . . . . . . . . . . . 17  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  <-> 
w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) ) )
20 eleq1 2356 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  ->  (
w  e.  ( A  .P.  ( B  +P.  C ) )  <->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( A  .P.  ( B  +P.  C ) ) ) )
2119, 20syl6bi 219 . . . . . . . . . . . . . . . 16  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  ->  ( w  e.  ( A  .P.  ( B  +P.  C ) )  <-> 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( A  .P.  ( B  +P.  C ) ) ) ) )
2221imp 418 . . . . . . . . . . . . . . 15  |-  ( ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  /\  w  =  ( v  +Q  u
) )  ->  (
w  e.  ( A  .P.  ( B  +P.  C ) )  <->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( A  .P.  ( B  +P.  C ) ) ) )
2317, 22syl5ibrcom 213 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  A  /\  y  e.  B )  /\  (
f  e.  A  /\  z  e.  C )
) )  ->  (
( ( v  =  ( x  .Q  y
)  /\  u  =  ( f  .Q  z
) )  /\  w  =  ( v  +Q  u ) )  ->  w  e.  ( A  .P.  ( B  +P.  C
) ) ) )
2423exp4b 590 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( ( x  e.  A  /\  y  e.  B )  /\  (
f  e.  A  /\  z  e.  C )
)  ->  ( (
v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
2524com3l 75 . . . . . . . . . . . 12  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( f  e.  A  /\  z  e.  C ) )  -> 
( ( v  =  ( x  .Q  y
)  /\  u  =  ( f  .Q  z
) )  ->  (
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
2625exp4b 590 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( ( f  e.  A  /\  z  e.  C )  ->  (
v  =  ( x  .Q  y )  -> 
( u  =  ( f  .Q  z )  ->  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  ->  (
w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C
) ) ) ) ) ) ) )
2726com23 72 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( v  =  ( x  .Q  y )  ->  ( ( f  e.  A  /\  z  e.  C )  ->  (
u  =  ( f  .Q  z )  -> 
( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u
)  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) ) )
2827rexlimivv 2685 . . . . . . . . 9  |-  ( E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y )  ->  (
( f  e.  A  /\  z  e.  C
)  ->  ( u  =  ( f  .Q  z )  ->  (
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) )
2928rexlimdvv 2686 . . . . . . . 8  |-  ( E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y )  ->  ( E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z )  ->  (
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
3029com3r 73 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y )  ->  ( E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z )  ->  (
w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C
) ) ) ) ) )
3116, 30sylbid 206 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
v  e.  ( A  .P.  B )  -> 
( E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z )  ->  ( w  =  ( v  +Q  u
)  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
3231imp3a 420 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( A  .P.  B )  /\  E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z ) )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) )
3313, 32sylbid 206 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( A  .P.  B )  /\  u  e.  ( A  .P.  C ) )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) )
3433rexlimdvv 2686 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. v  e.  ( A  .P.  B ) E. u  e.  ( A  .P.  C ) w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) )
358, 34sylbid 206 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( ( A  .P.  B )  +P.  ( A  .P.  C ) )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) )
3635ssrdv 3198 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A  .P.  B
)  +P.  ( A  .P.  C ) )  C_  ( A  .P.  ( B  +P.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165  (class class class)co 5874    +Q cplq 8493    .Q cmq 8494   P.cnp 8497    +P. cpp 8499    .P. cmp 8500
This theorem is referenced by:  distrpr  8668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-plp 8623  df-mp 8624
  Copyright terms: Public domain W3C validator